Given a tree $(T,\sqsubset)$, a map $\varphi: T\longrightarrow \mathbb{Q}$ is order preserving if and only if $\varphi(x)<\varphi(y)$ iff $x\sqsubset y$. The forcing notion $Fn^{op}(T,\mathbb{Q})$ of all finite partial function from $T$ on the rationals which are order preserving is ccc and adds a order preserving map from $T$ on the rationals in the generic extension.
On the sequel denote by $\mathcal{A}$ the class of all Aronszajn trees. My question is the following. Why the forcing notion $\prod^{fin}_{T\in\mathcal{A}} Fn^{op}(T,\mathbb{Q})$ is ccc? To see this my strategy was to prove that for every $\mathcal{I}\in [\mathcal{A}]^{<\omega}$ the product $\prod_{T\in\mathcal{I}} Fn^{op}(T,\mathbb{Q})$ is ccc. I also know that the tree product $T'=\oplus_{T\in\mathcal{I}} T$ is Aronszajn and hence $Fn^{op}(T',\mathbb{Q})$ is ccc but, is there any relation between the posets $Fn^{op}(T',\mathbb{Q})$ and $\prod^{fin}_{T\in\mathcal{A}} Fn^{op}(T,\mathbb{Q})$ which allows to see that the last one is ccc?
Every comment would be appreciated.